Question

The distributions of X and of Y are described here. If X and Y are independent, determine the joint probability distribution of X and Y.

Answer 1

To determine the joint probability distribution of X and Y, we need to know their individual probability distributions and if they are independent.

Step-by-step explanation:

The joint probability distribution of X and Y can be determined if we know the individual probability distributions of X and Y and if X and Y are independent.

Let's assume that X can take on values x1, x2, ..., xn with probabilities p(X=x1), p(X=x2), ..., p(X=xn) respectively. Similarly, let's assume that Y can take on values y1, y2, ..., ym with probabilities p(Y=y1), p(Y=y2), ..., p(Y=ym).

If X and Y are independent, then the joint probability of X and Y is given by multiplying the individual probabilities. Therefore, the joint probability distribution table for X and Y would include all possible combinations of values for X and Y along with their corresponding probabilities.

Answer 2

The joint probability distribution of X and Y is shown below.

Step-by-step explanation:

The distributions of X and of Y are described as follows:

X : 0 1

P (X) : 0.23 0.77

Y : 1 2 3

P (Y) : 0.40 0.22 0.38

It is provided that X and Y are independent.

That is:

P (X ∩ Y) = P (X) × P (Y)

Compute the joint probability distribution of X and Y as follows:

`P(X=0,Y=1)=P(X=0)* P(Y=1)=0.23* 0.40=0.92\\\\P(X=0,Y=2)=P(X=0)* P(Y=2)=0.23* 0.22=0.0506\\\\P(X=0,Y=3)=P(X=0)* P(Y=3)=0.23* 0.38=0.0874\\\\P(X=1,Y=1)=P(X=1)* P(Y=1)=0.77* 0.40=0.308\\\\P(X=1,Y=2)=P(X=1)* P(Y=2)=0.77* 0.22=0.1694\\\\P(X=1,Y=3)=P(X=1)* P(Y=3)=0.77* 0.38=0.2926`

X 0 1

Y

1 0.9200 0.3080

2 0.0506 0.1694

3 0.0874 0.2926